The concept of limits in mathematics is foundational, particularly in calculus. A limit describes the value that a function (or sequence) approaches as the input (or index) approaches a specific value.
Formal Definition
If
𝑓
(
𝑥
)
f(x) is a function and
𝑐
c is a value, then:
lim
⁡
𝑥
→
𝑐
𝑓
(
𝑥
)
=
𝐿
x→c
lim
​
f(x)=L
means that as
𝑥
x gets arbitrarily close to
𝑐
c,
𝑓
(
𝑥
)
f(x) approaches the value
𝐿
L.
Key Types of Limits
Finite Limits as
𝑥
→
𝑐
x→c: The value
𝑓
(
𝑥
)
f(x) approaches as
𝑥
x approaches
𝑐
c. Example:
lim
⁡
𝑥
→
2
(
3
𝑥
+
1
)
=
7
x→3
lim
​
(3x+1)=10
Limits at Infinity: When
𝑥
x grows large positively or negatively. Example:
lim
⁡
𝑥
→
∞
1
𝑥
=
0
x→∞
lim
​
x
1
​
=0
One-sided Limits:
Left-hand limit (
𝑥
→
𝑐
−
x→c
−
):
𝑥
x approaches
𝑐
c from the left.
Right-hand limit (
𝑥
→
𝑐
+
x→c
+
):
𝑥
x approaches
𝑐
c from the right. Example:
lim
⁡
𝑥
→
1
−
∣
𝑥
−
1
∣
=
0
and
lim
⁡
𝑥
→
1
+
∣
𝑥
−
1
∣
=
0
x→1
−
lim
​
∣x−1∣=0and
x→1
+
lim
​
∣x−1∣=0
Limits Involving Infinity:
𝑓
(
𝑥
)
f(x) can tend to infinity or negative infinity. Example:
lim
⁡
𝑥
→
0
+
1
𝑥
=
∞
x→0
+
lim
​
x
1
​
=∞
Evaluating Limits
Direct Substitution: If
𝑓
(
𝑐
)
f(c) is defined,
lim
⁡
𝑥
→
𝑐
𝑓
(
𝑥
)
=
𝑓
(
𝑐
)
limt
x→c
​
f(x)=f(c).
Factoring: Simplify the expression by factoring.
Rationalizing: Used for functions involving roots.
L ' Hopital 's Rule: For indeterminate forms like
0
0
0
0
​
or
∞
∞
∞
∞
​
, differentiate numerator and denominator. Example:
limt
⁡
𝑥
→
0
sin
⁡
𝑥
𝑥
=
1
x→0
limt
​
x
sinx
​
=1
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